340 XV NORMED ALGEBRAS AND SPECTRAL THEORY (15.4.15) Let A be a commutative star algebra with unit element e^Q. Suppose that there exists x0 e A such that the subalgebra generated by e, x0 and x* is dense in A. Then the mapping x*-*x(*o) is a homeomorphism o/X(A) onto SpA(x0)- As in (15.3.6) we see that A is separable, and that it is enough to prove that the relation XI(XQ) = X2(*o) for two characters &, %2 implies that Xi = Xf>. Now, from (15.4.14) we deduce that Xi(*o) = tofro) and thence that Xi(P(xo,x$) = X2(*(x<»x$) for a11 polynomials PeC[X,Y]. Since by hypothesis the P(*0 > **) are dense in A, and since %19 I2 are continuous on A, the result follows. PROBLEMS 1. Let A be a normed algebra endowed with a continuous involution x\->x*. If ||#||! = sup(||*||, ||jc*||), show that with the norm \\x\\i the algebra A becomes a normed algebra with involution, and that the norms ||x|| and \\x\\ i are equivalent. 2. Let A be a commutative Banach algebra without radical (Section 15.2, Problem 7). Show that every involution on A is continuous (use Problem 20 of Section 15.3). 3. Let A be a commutative Banach algebra with involution, having a unit element. A character % of A is said to be hermitian if xC**) = x(*) f°r aU xe A. Show that every character of A is hermitian if and only if, for each x e A, the element 1 + xx* is invertible in A. (If every character of A is hermitian, then #(1 H- xx*) is a function which is everywhere >0 on X(A); use (15.3.4). If a character x of A is not hermitian, then there exists a self-adjoint y e A such that xOO = '» whence %(1 + yy*) ~ 0, and 1 + yy* is not invertible.) Hence give an example of a commutative Banach algebra with involution, having a unit element, for which there exist nonhermitian characters. 4. (a) If A is a star algebra with unit element, then ||*||2 =p(x*x) for all x e A (prove this first when x is self-adjoint). Deduce that A is without radical. oo (b) Let A be the algebra of Problem 2 of Section 15.1. If x = £ £nTn, define x* to be n = 0 oo , 2 snT". This defines an involution on A for which A is a commutative Banach algebra n = 0 with involution, having a unit element, and such that the only character of A (Section 15.3, Problem 2) is hermitian. 5. With respect to the involution/*(/)=/(—/) on the Beurling algebra (Section 15.1, Problem 4 and Section 15.2, Problem 6), show that all the characters are hermitian. 6. On the algebra A of Problem 7 of Section 15.3, consider the involution induced by /i-*/, and extend this involution to A by putting e* = e. Then A is a Banach algebra with involution. Find its hermitian characters. Problem 7) is